Can an Agent persona go into a Host while the decker's persona stays on the grid?
An Agent is an autonomous program (SR5 page 246), and entering a host is a Matrix Action (SR5 page 239) that an Agent program is capable of performing. The only requirement is that the Agent must have at least one MARK on the host, and MARKs can't be shared, so it would have to obtain said MARK on it's own. This is potentially problematic, especially against higher rating hosts, but not impossible, so the answer to this question is "Yes, absolutely."
If the decker's persona is in silent mode, are the Agents in silent mode? Can an Agent be in silent mode while the decker's persona is not?
Not necessarily, and absolutely, respectively.
First of all, a decker can run an Agent program on a commlink and his own persona on a deck thus separating the two devices entirely for the purposes of Running Silently.
Secondly, when someone uses a device to access the Matrix that device icon is subsumed by the Persona icon (SR5 page 235), but an Agent running on the same device has it's own icon in turn. Since Running Silently can be toggled for on a per device or persona basis (SR5 page 235) you can absolutely have an Agent run silently while the decker doesn't, or vice versa.
Finally, fseperent's comment about having a single agent or persona from a single device is not entirely correct; while it's true that any one
person can only have a single persona at a time (SR5 p235) it may or may not be possible to run multiple agents on a single device, depending on your interpretation of the rules, and it's definitively possible to run two personas on a single device by virtue of having an agent and a person using the same device.
The multiple agent on a single device topic has come up several times, and it really boils down to how one interprets the statement that agents are programs, and one cannot run the same program more than once on a single device to gain the benefit twice. The argument usually comes down to whether two Agent programs are the same as two Edit programs, for example, or if a Rating 3 agent program is the same as a Rating 4 agent program.
All I can say is that at my table we allow multiple agent programs, even of the same rating, to run simultaneously on a single device. The significant up-front investment cost (agents are expensive, after all) along with the increased risk of both exposure and damage (each additional agent is one more icon that can be spotted and/or attacked, and all damage is taken by the device they're running on) balance the benefits of running multiple agents. This is obviously true for my table and it doesn't have to be for others, and the rules here are ambiguous enough that multiple interpretations are possible.